On closedness and simple connectedness of adjoint and coadjoint orbits (Q1320363)

Let \(\mathfrak g\) be a finite dimensional real Lie algebra and \(\mathfrak h\) a subalgebra of \(\mathfrak g\). For a subset \(S\) of a group we write \(\langle S\rangle\) for the subgroup generated by \(S\). We define \(\text{Inn}_{\mathfrak g}({\mathfrak h}) := \langle e^{\text{ad}{\mathfrak h}}\rangle\) and \(\text{INN}_{\mathfrak g}({\mathfrak h}) := \overline{\text{Inn}_{\mathfrak g}({\mathfrak h})}\). We also set \(\text{Inn}_{\mathfrak g} := \text{Inn}_{\mathfrak g}({\mathfrak g})\) and \(\text{INN}_{\mathfrak g} := \text{INN}_{\mathfrak g}({\mathfrak g})\). Then the adjoint action is the action of \(\text{Inn}_{\mathfrak g}\) on the Lie algebra \(\mathfrak g\) and the coadjoint action is the action of \(\text{Inn}_{\mathfrak g}\) on \({\mathfrak g}^*\) via \(g.v := (g^{-1})^*(v) = v \circ g^{-1}\). The main results of this paper are: (1) Every coadjoint orbit of strict convexity type is closed and simply connected; (2) Every adjoint orbit meeting a Cartan subalgebra \(\mathfrak t\) and every coadjoint orbit meeting \({\mathfrak t}^*\) is closed and one component of its Zariski closure.